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Lipics-ITCS-2019-29.Pdf (0.5 From Local to Robust Testing via Agreement Testing Irit Dinur1 Department of Mathematics and Computer Science, Weizmann Institute of Science, Rehovot, Israel [email protected] Prahladh Harsha2 Tata Institute of Fundamental Research, India [email protected] Tali Kaufman Department of Computer Science, Bar-Ilan University, Ramat Gan, Israel [email protected] Noga Ron-Zewi Department of Computer Science, University of Haifa, Haifa, Israel [email protected] Abstract A local tester for an error-correcting code is a probabilistic procedure that queries a small subset of coordinates, accepts codewords with probability one, and rejects non-codewords with probability proportional to their distance from the code. The local tester is robust if for non-codewords it satisfies the stronger property that the average distance of local views from accepting views is proportional to the distance from the code. Robust testing is an important component in constructions of locally testable codes and probabilistically checkable proofs as it allows for composition of local tests. In this work we show that for certain codes, any (natural) local tester can be converted to a roubst tester with roughly the same number of queries. Our result holds for the class of affine- invariant lifted codes which is a broad class of codes that includes Reed-Muller codes, as well as recent constructions of high-rate locally testable codes (Guo, Kopparty, and Sudan, ITCS 2013). Instantiating this with known local testing results for lifted codes gives a more direct proof that improves some of the parameters of the main result of Guo, Haramaty, and Sudan (FOCS 2015), showing robustness of lifted codes. To obtain the above transformation we relate the notions of local testing and robust testing to the notion of agreement testing that attempts to find out whether valid partial assignments can be stitched together to a global codeword. We first show that agreement testing implies robust testing, and then show that local testing implies agreement testing. Our proof is combinatorial, and is based on expansion / sampling properties of the collection of local views of local testers. Thus, it immediately applies to local testers of lifted codes that query random affine subspaces m in Fq , and moreover seems amenable to extension to other families of locally testable codes with expanding families of local views. 2012 ACM Subject Classification Theory of computation → Computational complexity and cryptography 1 Supported by ERC-CoG grant number 772839. 2 Research supported in part by the UGC-ISF grant and the Swarnajayanti Fellowship. Part of the work was done when the author was visiting the Weizmann Institute of Science. © Irit Dinur, Prahladh Harsha, Tal Kaufman, and Noga Ron-Zewi; licensed under Creative Commons License CC-BY 10th Innovations in Theoretical Computer Science (ITCS 2019). Editor: Avrim Blum; Article No. 29; pp. 29:1–29:18 Leibniz International Proceedings in Informatics Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany 29:2 From Local to Robust Testing via Agreement Testing Keywords and phrases Local testing, Robust testing, Agreement testing, Affine-invariant codes, Lifted codes Digital Object Identifier 10.4230/LIPIcs.ITCS.2019.29 1 Introduction Our main result shows a transformation from local testing to robust testing for the class of affine-invariant lifted codes. We start by describing the notions of local testing, robust testing, and lifted codes. 1.1 Local testing and robust testing A code is a subset C ⊆ Σn. The elements of C are called codewords, Σ is the alphabet of the code, and n is the block length. The rate of the code is the ratio (log|Σ| |C|)/n. The code is linear if Σ = Fq where Fq is the finite field of q elements, and C is an Fq-linear n subspace of Fq . It will be convenient to think of codewords in C as functions f : U → Σ where U is a domain of size n. For a pair of functions f, g : U → Σ we let dist(f, g) denote the fraction of inputs x ∈ U for which f(x) 6= g(x). The relative distance dist(C) of the code is the minimum of dist(f, g) over all codewords f, g ∈ C. For a function f : U → Σ we let dist(f, C) denote the minimum of dist(f, g) over all codewords g ∈ C. A local tester for the code C is a probabilistic oracle algorithm that on oracle access to a function f : U → Σ makes at most Q queries to f, and accepts f ∈ C with probability one, while rejecting f 6∈ C with probability at least α · dist(f, C). We refer to Q as the query complexity of the tester, and to α as the soundness. In this work we shall restrict our attention to local testers that pick a random subset K ⊆ U of cardinality Q according 3 to some distribution, and accept if and only if f|K ∈ C|K . The requirement then is that f|K ∈ C|K with probability one whenever f ∈ C, and Pr[f|K ∈/ C|K ] ≥ α · dist(f, C) (1) K otherwise. In this work we will be interested in the stronger notion of robustness. We say that a local tester as above is robust if for non-codewords the average distance of its local views from accepting views is proportional to the distance of the given function from the code. That is, as before we require that f|K ∈ C|K with probability one whenever f ∈ C, but instead of (1) we now require that EK dist(f|K ,C|K ) ≥ α · dist(f, C) (2) whenever f∈ / C. Here we refer to α as the robustness of the tester. The notion of robustness was introduced by Ben-Sasson and Sudan [8] based on analogous notions for probabilistically checkable proofs [5, 15]. Robustness is a natural property of local testers that relates the global distance of a function from the code to its local distance from accepting views on local views. Moreover, robustness is also an important ingredient in constructions of locally testable codes and probabilistically checkable proofs as it allows for composition of local tests. Specifically, it follows by definition that if a code C is robustly 3 Local testers may generally apply a more complex predicate on f|K . However, natural local testers are typically of the restricted form we consider, and moreover it can be shown that a local tester for a linear code must be of this form [6]. I. Dinur, P. Harsha, T. Kaufman, and N. Ron-Zewi 29:3 testable with query complexity Q and soundness α, and additionally each local restriction 0 0 C|K is locally testable with query complexity Q and soundness α , then the code C is locally testable with query complexity Q0 and soundness α · α0. This property is useful when local restrictions can be tested efficiently which can happen if the code has many symmetries (as is the case with the class of lifted codes considered in this work) or can be achieved, in the case of probabilistically checkable proof, by attaching a short proof of proximity. One can easily observe that (2) implies (1) since f|K ∈/ C|K whenever dist(f|K ,C|K ) > 0, so robustness is a stronger requirement than local testing. For the other direction, note that a local tester with soundness α has robustness at least α/Q since dist(f|K ,C|K ) ≥ 1/Q whenever f|K ∈/ C|K . A natural question is whether this loss in roubstness is necessary, and whether robustness is strictly stronger notion than local testing. In this work we shall show that this loss is unnecessary for the class of lifted codes, discussed below. 1.2 Lifted codes ` Lifted codes are specified by a base code C ⊆ {Fq → Fq} and a dimension m ≥ `. We further assume that the base code C is linear and affine-invariant, that is, for any codeword f ∈ C, ` ` and for any affine transformation A : Fq → Fq it holds that f ◦ A ∈ C. Given these we define `%m m the lifted code C to be the code consisting of all functions f : Fq → Fq that satisfy that f|L ∈ C for any `-dimensional affine subspace L. Lifted codes were first introduced by Ben-Sasson et al [7], and their local testability properties were further explored in subsequent work [20, 21, 19]. They are a natural generalization of the well-studied family of Reed-Muller codes, and moreover they also give rise to new families of locally testable codes that outperform Reed-Muller codes in certain range of parameters [20]. Specifically, lifted codes lead to one of the two known constructions (the other one being tensor codes [8, 9, 27, 24]) of high-rate locally testable codes (i.e., locally testable codes with rate approaching one and sublinear locality). Generally, lifted codes form a natural subclass of affine-invariant codes satisfying the “single-orbit characterization” property that is known to imply local testability, as well as local decodability [23]. There is a natural local test associated with lifted codes: on oracle access to a function m f : Fq → Fq, pick a uniform random `-dimensional affine subspace L and accept if and only if f|L ∈ C. It follows immediately by definition that this test accepts any valid codeword f ∈ C`%m with probability one, but more work is required to show that this test is sound. Specifically, since the test forms a single orbit characterization, it follows from [23] that it has soundness roughly q−2`. The dependence of the soundness on the dimension ` was later eliminated in [21] who showed soundness that is only a function of q (though an extremely quickly decaying one).
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